JF:
As you mention, there were lots of military related periodicals published in the UK in the early to late 19th Century with similar titles, with the United Services Gazette being one of them. Brief history of Gazette here for those interested:
link
As Andrew mentions, I think I've seen issues of the Gazette on Goggle Books or the Internet Archive (probably through links provided to me by Andrew, thanks BTW) but I haven't saved any of them and because of the similarity of names of the various gazettes and journals I am not certain that what I saw was the "United Services Gazette" and not, instead for example, the "Naval and Military Gazette and Weekly Chronicle of the United Service."
Making the same searches you have probably already made, I've not been able to find any copies of the "United Services Gazette" available on the web so far-if I do I'll post links here. I did find dozens of secondary references to the United Services Gazette so I'm just a bit surprised that I was not able to find any copies available online.
What I have found so far (which I provide although it's not what you requested because it might be of some use to you or others) is:
First of all, for India related stuff, is the web site for Families in British India-a sort of genealogy site. The web site has a list of military related periodicals here:
link
The British Newspaper archive, a fee service, has the Naval and Military Gazette and Weekly Chronicle of the United Service available here:
link
And finally, for now at least, the Hathni Trust has individual issues of the United Service Journal and Naval and Military Magazine available here:
link
As Andrew suggests, you could try Archive dot Org. I tried and got no useful results, but I always have problems with its search function and so my results may not be typical.
Anyway, good luck in your search. If we can be of further help don't hesitate to ask. And if you do find a source for the United Services Gazette then I, and probably many others, would appreciate learning of it.
Dennis